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45 minutes is ideal, in my opinion. I consider myself a pretty advanced singer, and I feel that, although after I'm still able to sing or do things after 45 minutes , the efficiency is worse. 30 minutes can be very strong, but it won't be a relaxed lesson, and it's therefore impersonal for starters. Then, the last 5 minutes of a 60 minute lesson is usually spoiled.

Obviously it depends from teacher to teacher what you'll get, but usually you'll start with a warm-up singing scales and runs as the first part and singing a song by the teacher or chosen by you as the second part.

That which is more important: the regularity. I suggest you take weekly lessons. 30-minute weekly lessons will be a lot better than 45-minute/60-minute lessons 2 times a month, especially for starters as it's not important how much you learn, but how much you repeat/practice what you've learned. Master one thing at a time.

Right now I loss a lot of interest not motivated in learning vocal and piano not much at home practising as I am reaching 30 years old.

I am under regular medication to treat depression from psychiatrist and she had added me new medication since last two months ago as I was very stress in work as I express to her in my follow up appointment.

So is 45 minutes really ideal for me once a week lesson no need to go for an hour once a week lesson in vocal?

If you have the money, go for the hour lessons. It's just that I don't find there's that much of a difference between the two. You do have more space in a 60 minute lesson, it's more relaxed. I was just trying to say that you have enough to learn in a 45 minute lesson. Just make sure you'll take weekly lessons if you want to improve steadily.

Singing has everything to do with confidence. Make a decision and go on by that decision. This hesitation you seem to have is only slowing you down. You seem to prefer 60 minutes lessons, well do those, then! Go for it.